Model Theory, Combinatorics and Valued fields

Model Theory, Combinatorics and Valued fields
A trimester at the Institut Henri Poincaré, 8 January - 6 April 2018

Organizing Committee
Elisabeth Bouscaren (CNRS - Orsay)
Zoé Chatzidakis (CNRS - ENS)
Martin Hils (Münster)
Dugald Macpherson (Leeds)
Frank Wagner (Lyon 1)

Scientific Committee
Elisabeth Bouscaren (CNRS - Orsay)
Emmanuel Breuillard (Münster/Orsay)
Ehud Hrushovski (Oxford)
François Loeser (UPMC)
Anand Pillay (Notre Dame)
Tom Scanlon (UC Berkeley)

NEW: Videos of the trimester: presentation and the three workshops.

In order to participate in the programme, you must register on the IHP web site.

Information about courses.
Do not forget to look at the agenda to check rooms.

Download the poster: pdf or jpg

Aims of the programme

Model theory is a branch of mathematical logic which deals with the relationship between formal logical languages (e.g. first order logic, or variants such as continuous logic) and mathematical objects (e.g. groups, or Banach spaces). It analyses mathematical structures through the properties of the category of its definable sets. Significant early applications of model theory include Tarski's decidability results in the 1920s (algebraically closed fields, real closed fields), and in the 1960s the well-known Ax-Kochen/Ershov results on the model theory of Henselian valued fields.

These last few years have seen an extremely rapid development of the powerful tools introduced for stable structures in a much larger context, that of “tame” structures. Our main themes for this programme aim to develop both the internal model theory of tame structures and their recent applications.

The programme will bring together researchers on the following topics:

(i) Model theory and application to combinatorics. Additive combinatorics (approximate subgroups and variations); Around Szemerédi Regularity Lemma and Density Theorem; Pseudofinite structures (e.g., ultraproducts of finite structures); Vapnik-Chervonenkis theory, applications, and NIP theories; Continuous model theory; Generalised stability theory and tame structures.
(ii) Model theory of valued fields and applications. The prime focus is on the model theory of Henselian valued fields with the valuation topology, often with extra structure and under assumptions which ensure that definable sets can be understood. Motivic integration; Algebraically closed valued fields, imaginaries, and Berkovich spaces; Valued fields with additional structure; Transseries and surreal numbers; Definable groups.
(iii) Applications of model theory in geometry, analysis and number theory. Study of fields with operators and their applications to concrete problems; Applications of the Pila-Wilkie counting theorem.

The emphasis will be on the first two themes, where interactions and collaborations are still at an early stage. Theme (iii) is already well developed, and has connections with both themes (i) and (ii), mainly through concrete algebraic examples. While very present in the programme, it is less central.

We intend to concentrate the activities of theme (i) (around Combinatorics) in the period 15 January - 9 February, and those of theme (ii) (around valued fields) in the period 12 February - 9 March, leading up to and including the second meeting. The third and final meeting will be general, including all three themes. However, we expect to have people from all themes of the programme at any point of time.

Structure of the trimester

In addition to weekly seminars and courses, the trimester will have four meetings:

Model theory, combinatorics and valued fields, CIRM, 8 - 12 January 2018. Introductory preschool of the trimester.

Model theory and combinatorics, 29 January - 2 February 2018. IHP.

Model theory of valued fields, 5 - 9 March 2018. IHP.

Model theory and applications, 26 - 30 March 2018. IHP.